Empty Set is Open in Metric Space

Theorem

Let $M = \struct {A, d}$ be a metric space.

Then the empty set $\O$ is an open set of $M$.

Proof

By definition, an open set $S \subseteq A$ is one where every point inside it is an element of an open ball contained entirely within that set.

That is, there are no points in $S$ which have an open ball some of whose elements are not in $S$.

As there are no elements in $\O$, the result follows vacuously.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Theorem $6.4$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.10$